Distribution of periodic orbits of symbolic and axiom A flows.

*(English)*Zbl 0637.58013Let A be an \(n\times n\) 0-1 matrix. For given matrix A, \(\Sigma_{\hat A}\) denotes the set of all doubly infinite sequences and \(C(\Sigma_ A)\) the space of continuous complex-valued functions on \(\Sigma_ A\). For \(f\in C(\Sigma_{\hat A})\) let \((\Sigma^ f_{\hat A},\sigma^ f)\) denote the symbolic flow, defined on the suspension space \(\Sigma^ f_ A\) by usual manner. Let G be a bounded Borel measurable real-valued function on \(\Sigma^ f_ A\). Periodic orbits of \((\Sigma^ f_ A,\sigma^ f)\) are denoted by \(\tau\), \(\tau\) (G) denotes the integral of G, with respect to time, over one period for \(\tau\). Thus \(\tau\) (1) is the period of \(\tau\), and \(\tau\) (G)/\(\tau\) (1) is the mean value of G over \(\tau\). Define measure \(Q^ f\) on \({\mathbb{R}}^+\) by \(Q^ f(I)=\#\{\tau: \tau (1)\in I\}\). The main results of this paper concern the asymptotic behavior of this measure, when the flow is weakly mixing. Also further analogous measures are constructed and investigated.

Reviewer: A.Klíč

##### Keywords:

subshift of finite type; invariant measure; symbolic flow; suspension space; asymptotic behavior; weakly mixing
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